Best Known (56, 65, s)-Nets in Base 4
(56, 65, 262149)-Net over F4 — Constructive and digital
Digital (56, 65, 262149)-net over F4, using
- 41 times duplication [i] based on digital (55, 64, 262149)-net over F4, using
- net defined by OOA [i] based on linear OOA(464, 262149, F4, 9, 9) (dual of [(262149, 9), 2359277, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(464, 1048597, F4, 9) (dual of [1048597, 1048533, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(461, 1048576, F4, 9) (dual of [1048576, 1048515, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(441, 1048576, F4, 6) (dual of [1048576, 1048535, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(464, 1048597, F4, 9) (dual of [1048597, 1048533, 10]-code), using
- net defined by OOA [i] based on linear OOA(464, 262149, F4, 9, 9) (dual of [(262149, 9), 2359277, 10]-NRT-code), using
(56, 65, 524300)-Net over F4 — Digital
Digital (56, 65, 524300)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(465, 524300, F4, 2, 9) (dual of [(524300, 2), 1048535, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(465, 1048600, F4, 9) (dual of [1048600, 1048535, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(461, 1048576, F4, 9) (dual of [1048576, 1048515, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(441, 1048576, F4, 6) (dual of [1048576, 1048535, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(44, 24, F4, 2) (dual of [24, 20, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(44, 85, F4, 2) (dual of [85, 81, 3]-code), using
- Hamming code H(4,4) [i]
- discarding factors / shortening the dual code based on linear OA(44, 85, F4, 2) (dual of [85, 81, 3]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- OOA 2-folding [i] based on linear OA(465, 1048600, F4, 9) (dual of [1048600, 1048535, 10]-code), using
(56, 65, large)-Net in Base 4 — Upper bound on s
There is no (56, 65, large)-net in base 4, because
- 7 times m-reduction [i] would yield (56, 58, large)-net in base 4, but